ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A small residual can hide a large stationary error

Example

For $0 < ε ≪ 1$ , consider the two-state column-stochastic matrix

$P_{ε} = (\begin{array}{cc} 1 - ε & ε \\ ε & 1 - ε \end{array}) .$

It has stationary law $π = (1 / 2, 1 / 2)^{T}$ . Take $ε = 1 / 100$ and candidate $x = (3 / 4, 1 / 4)^{T}$ . The residual is small, while the stationary error is $1 / 2$ . The exact Dobrushin certificate is sharp and exposes the large inverse factor.

Verification

Given: The displayed matrix with $ε = 1 / 100$ and the candidate vector $x = (3 / 4, 1 / 4)^{T}$ .

[F1] Column-overlap formula for the zero-sum ℓ1 coefficient: $δ (P) = 1 - \min_{j, k} \sum_{i} \min (P_{i j}, P_{i k})$ for a column-stochastic matrix.

[F2] m-step Dobrushin certificate for a stationary probability: if $q = δ (P^{m}) < 1$ , then $∥ x - π ∥_{1} \leq m ∥ P x - x ∥_{1} / (1 - q)$ .

[F3] Correct and certify a finite-chain stationary candidate: when the raw input is the nonnegative normalized vector $y = x$ , correction distance is zero; for exact rational P and y, the algorithm returns the residual, overlap coefficient, and bound when that coefficient is below one.

Proof technique: direct exact calculation.

1.1givenF3algebra

Direct multiplication gives $P π = π$ for $π = (1 / 2, 1 / 2)^{T}$ . Use the raw algorithm input $y = x = (3 / 4, 1 / 4)^{T}$ ; it is nonnegative and normalized, so its correction distance is $c = 0$ .

2.1givenstep 1.1algebra

The first coordinate of $P x$ is $(99 / 100) (3 / 4) + (1 / 100) (1 / 4) = 149 / 200$ , and the second is $51 / 200$ . Hence $r = P x - x = (- 1 / 200, 1 / 200)^{T}$ and $∥ r ∥_{1} = 1 / 100$ , while $∥ x - π ∥_{1} = 1 / 2$ .

3.1F1F2step 2.1algebra∎

The columns are $(99 / 100, 1 / 100)^{T}$ and $(1 / 100, 99 / 100)^{T}$ ; their overlap is $1 / 50$ . Thus [F1] gives $q = δ (P) = 49 / 50 < 1$ . With $m = 1$ , [F2] gives the exact bound $(1 / 100) / (1 / 50) = 1 / 2$ , equal to the actual error.

Depends on

Used by

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Sources

Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, Chapter 11 (standard reference, not scraped)
Ilse C. F. Ipsen and Teresa M. Selee, Ergodicity Coefficients Defined by Vector Norms (standard reference, not scraped)